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Solve the following quantitative compari...

Solve the following quantitative comparison problem by plugging in the number 0,1,2, -2, and `1//2` in that order - when possible.
`{:("Column A"," ","Column B"),(ab^2," ",a^2b):}`

A

If column A is larger

B

If column B is larger

C

If the columns are equal

D

If there is not enough information to decide

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will evaluate the expressions in Column A and Column B by plugging in the numbers 0, 1, 2, -2, and 1/2 in that order. ### Step-by-Step Solution: 1. **Plug in A = 0, B = any value (let's say B = 1 for simplicity)**: - Column A: \( AB^2 = 0 \cdot 1^2 = 0 \) - Column B: \( A^2B = 0^2 \cdot 1 = 0 \) - **Conclusion**: Column A = Column B (both are 0). 2. **Plug in A = 1, B = 2**: - Column A: \( AB^2 = 1 \cdot 2^2 = 1 \cdot 4 = 4 \) - Column B: \( A^2B = 1^2 \cdot 2 = 1 \cdot 2 = 2 \) - **Conclusion**: Column A (4) is greater than Column B (2). 3. **Plug in A = 2, B = 1**: - Column A: \( AB^2 = 2 \cdot 1^2 = 2 \cdot 1 = 2 \) - Column B: \( A^2B = 2^2 \cdot 1 = 4 \cdot 1 = 4 \) - **Conclusion**: Column A (2) is less than Column B (4). 4. **Plug in A = -2, B = 1**: - Column A: \( AB^2 = -2 \cdot 1^2 = -2 \cdot 1 = -2 \) - Column B: \( A^2B = (-2)^2 \cdot 1 = 4 \cdot 1 = 4 \) - **Conclusion**: Column A (-2) is less than Column B (4). 5. **Plug in A = 1/2, B = 2**: - Column A: \( AB^2 = \frac{1}{2} \cdot 2^2 = \frac{1}{2} \cdot 4 = 2 \) - Column B: \( A^2B = \left(\frac{1}{2}\right)^2 \cdot 2 = \frac{1}{4} \cdot 2 = \frac{1}{2} \) - **Conclusion**: Column A (2) is greater than Column B (1/2). ### Summary of Results: - For A = 0: Column A = Column B - For A = 1, B = 2: Column A > Column B - For A = 2, B = 1: Column A < Column B - For A = -2, B = 1: Column A < Column B - For A = 1/2, B = 2: Column A > Column B ### Final Conclusion: Since the results vary depending on the values plugged in, we conclude that there is not enough information to decide which column is consistently larger. Therefore, the correct answer is: **Option D: There is not enough information to decide.**
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